Multiobjective Optimization of Non-Smooth PDE-Constrained Problems — Switches, State Constraints and Model Order Reduction
In almost all technical applications, multiple criteria are of interest – both during development as well as operation. Examples are fast but energy efficient vehicles and constructions that have to be light as well as stable. The goal in the resulting multiobjective optimization problems is the computation of the set of optimal compromises – the so-called Pareto set. A decision maker can then select an appropriate solution from this set. In control applications, it is possible to quickly switch between different compromises as a reaction to changes in the external conditions.
The Pareto set generally consists of infinitely many compromise solutions, its numerical approximation is therefore considerably more expensive than the solution of scalar optimization problems. This can quickly result in prohibitively large computational cost, particularly in situations where solutions to the underlying system are computationally expensive. For instance, this is the case when the system is described by a partial differential equation (PDE). In this context, surrogate models that can be solved significantly faster than classical numerical approximations by the finite element method are frequently used. In the case of non-smooth PDEs, reducing the computational cost is particularly important since these problems are often significantly more expensive to solve than smooth problems. However, the surrogate models introduce an approximation error intro the system, which has to be quantified and considered both in the analysis and the development of numerical algorithms. For non-smooth problems, literature on this topic is currently scarce.
The goal of this project is the development of efficient numerical methods to solve multiobjective optimization problems that are constrained by non-smooth PDEs. In the first step, optimality conditions for the non-smooth PDE-constrained problems will be derived, and the (hierarchical) structure of the Pareto sets will be analyzed. Building on this, algorithms for the computation of Pareto sets will be developed for these problems. The methods will be used for the optimization of problems with max-terms, contact problems, and time dependent hybrid and switched systems. In order to handle the numerical effort, reduced order modeling techniques – such as Reduced Basis, Proper Orthogonal Decomposition, and more recent approaches based on the Koopman operator – will be extended to the non-smooth setting. This requires the consideration of inexactness in the convergence analysis. Finally, the algorithms will be applied to several different problem settings in cooperation with other members of the Priority Programme.